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Matrices-Matrix Algebra

**Basic Concepts of Matrices**

A rectangular array of numbers and functions in an ordered form is known as Matrix, where numbers and functions are known as components of the matrix.

Let us suppose you have 5 iPhones and 10 Samsung Tablets(You are a rich man!), you can express it as [5 10] knowing that the number inside represents iPhones and Tablets. Now I have 3 iPhones and 6 Samsung Tablets ( I am also rich but not as rich as you), I can also express it as [3 6]. Now, both of these data can be transformed and expressed as follows:

A = | 5 10 |

| 3 6 |

In this arrangement 5 10 represents the first row, 3 6 represents the second row, 5 3 represents the first column and 10 6 represents the second column. The first column represents a number of iPhones possessed by you and I and second column represent a number of Samsung tablets possessed by us.

The given example above is a matrix. Matrices are denoted by capital letters like in this case. The horizontal lines of elements represent the row of the matrix and vertical lines of elements represent columns of the matrix. Here A has 2 rows and 2 columns. This is also known as Square Matrix which you will study below.

**Order of Matrix: **

The matrix consists of rows and columns. In the above example we saw Matrix A has 2 rows and 2 columns, thus an order of matrix is 2*2. Generally, the matrix consisting of x rows and y columns is read out as a matrix of order m*n or m*n matrix.

**Generally, m*n matrix has following rectangular arrangement: **

X = [x_{ij} ]_{m*n} where 1=< i =< m, 1=< j =< n

**Types of Matrices: **

Column Matrix: Matrix with one column is known as Column Matrix. It is represented as X = [x_{ij} ]_{m*1},matrix of order m * 1.

Row Matrix?: Matrix with one row is known as Row Matrix. It is represented as X = [x_{ij }]_{1*n,} matrix of order 1 * n.

Square Matrix?: Matrix with an equal number of rows and columns is known as Square Matrix i.e. when rows m = columns n. It is represented as X = [x_{ij} ]_{m*m},matrix of order m.

Diagonal Matrix: When all non-diagonal elements of a matrix become zero, it is known as Diagonal Matrix. Matrix X = [x_{ij} ]_{m*m} is called a diagonal matrix if x_{ij} = 0 when i ≠ j.

Scalar Matrix: When diagonal elements of a diagonal matrix becomes equal, it is known as Scalar Matrix. Matrix X = [x_{ij} ]_{m*m} is called a scalar matrix if x_{ij} = 0, when i ≠ j, x_{ij }= k, when i = j, for constant k.

Identity Matrix : When diagonal elements are 1 and rest are all zero it is known as Identity Matrix. Matrix X = [x_{ij} ]_{m*m} is an identity matrix, if x_{ij} = 1 for i =j and if x_{ij} = 0 for i ≠ j.

Zero Matrix: When all elements of a matrix are zero it is known as zero matrix.

Note: Two matrices X = [x_{ij} ] and Y = [y_{ij} ] are equal if they are of same order and each element of X and Y i.e.x_{ij }= y_{ij} for all i = j.

**Operations on Matrices: **

Addition of Matrices: Matrices can only be added when they are of same order. Suppose we have matrix A & B, both of order 2*2. The sum of matrices A and B is denoted by C.

Thus sum of two matrices A = [a_{ij} ] and B = [b_{ij }] is defined as matrix C = [c_{ij} ]_{m*n} , where c_{ij} = a_{ij}+ b_{ij} for all values of i and j.

Subtraction of Matrices?: Matrices can only be subtracted when they are of same order. Difference of two matrices A = [a_{ij} ] and B = [b_{ij }] is defined as matrix C = [c_{ij} ]_{m*n} , where c_{ij }= a_{ij} - b_{ij} for all values of i and j.

**Properties of Addition of Matrices: **

1) Commutative Law: If matrices A and B are of same order, then A+B = B+A

2) Associative Law: If matrices A and B and C are of same order then (A + B) + C = A + (B +C)

**Multiplication of Matrices: **For multiplication of matrices number of columns in one matrix should be equal to the number of rows in another matrix i.e. number of columns in matrix X is equal to the number of rows in matrix Y. If X = [x_{ij} ]_{m*n} and Y = [y_{jk} ]_{ n*p} then product of two matrices X and Y is the matrix Z of order m*p i.e. Z = [z_{ik} ]_{m*p }

**Properties of Multiplication of Matrices: **

1) Associative Law: For matrices X, Y and Z, (XY)Z = X(YZ)

2) Distributive Law: For matrices X, Y and Z , X(Y+Z) = XY + XZ or (X+Y)Z = XZ + YZ

Transpose of a Matrix: For matrix A = [a_{ij }]_{m*n} ,the matrix obtained by interchanging the rows and columns of A is called transpose of A, denoted by AT .

**Importance of Matrices: **

Matrices are very useful in the field of science as well as find applications in practical problems of real life, thus it is a requisite concept. Considering it as an important tool in mathematics, it simplifies our work to great extent as compared to other methods used in mathematics. Operations and notations of matrix are used in electronic spreadsheets, electrical circuits(for example solving questions of Kirchoff’s laws), in field of optics and quantum mechanics, in areas of business and science for cost estimation, budgeting, projection of sales, for seismic survey in the field of geology, robotics and automation concept of matrices plays an active role. Matrices play a vital role in computer-based applications, it is used for the projection of 3D image into a 2D screen, thus generating motions which seem realistic.

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